Algebra & Functions · Grades 7–10
Exponents and Powers: Rules, Negative and Fractional
Short answer: An exponent says how many times to multiply a base by itself: 2^10 = 1024. Negative exponents mean reciprocals; fractional exponents mean roots.
Whole-number exponents
2 multiplied by itself ten times is 1024. The base is repeated; the exponent counts the repeats.
2^10
Answer1024
Step-by-step working 3 steps
- 1Write it as a power
2^10
- 2Repeated multiplication
2 is multiplied by itself 10 times.
- 3Result
2^10 = 1024
Zero and negative exponents
Any non-zero number to the power 0 is 1. A negative exponent flips the base: 2^−3 = 0.125.
2^(−3)
Answer0.125
Step-by-step working 4 steps
- 1Write it as a power
2^−3
- 2Multiply the base 3 times
2 × 2 × 2 = 8
- 3Negative exponent means reciprocal
2^−3 = 1 / 2^3 = 1 / 8
- 4Result
2^−3 = 0.125
Fractional exponents are roots
The exponent 1/2 means the square root: 9^(1/2) = 3.
9^(1/2)
Answer3
Step-by-step working 3 steps
- 1Write it as a power
9^(1/2)
- 2Fractional exponent = root and power
9^(1/2) = (2th root of 9)^1
The denominator is the root, the numerator is the power.
- 3Result
9^0.5 = 3
Rules to remember
- aᵐ · aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (ab)ⁿ = aⁿbⁿ
Common mistakes
Multiplying base and exponent
2³ is 2 × 2 × 2, not 2 × 3.
Misreading −2²
The exponent binds first, so −2² = −4; use (−2)² for 4.
Adding exponents of different bases
The product rule needs the same base.
Put it into practice. Try your own numbers in the Exponent Calculator and compare each step with the method above.
Frequently asked questions
What is 0⁰?
It is left undefined in most school maths because different rules give different answers.
